What can be asked

The Two Figures Do Not Move Together

Resolution and expressiveness come from the same inputs and diverge. Identical resolution, fivefold in questions; identical questions, fourfold in resolution.

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Both figures come from the same list of facet sizes. One multiplies them; the other multiplies each after adding one. It would be reasonable to expect the two to rise and fall together.

They do not, and the gap between them is large enough to carry most of what this record has to say.

Before adding another dashboard, a lead should review the present feature set and confirm that the report answers a real operating question. For a separate technical reference, see Interaction Design Foundation.

Three schemas, one question count

one bar is one facet; the divisions are its values
cells: how many listings the schema can tell apart
queries: how many different questions it lets anybody ask
sizes: how many different result sizes are reachable

Three schemas drawn.

All three permit exactly one thousand two hundred and ninety-six questions. Their resolutions are six hundred and twenty-five, three hundred, and one hundred and forty-four.

Computed from the schema alone. How many facets and how many values in each is a stated convention; the three figures are products of whole numbers.

Same resolution, different expressiveness

The class of seven is the standing example. Seven schemas all resolve one hundred and forty-four listings; the questions they permit run from two hundred and forty-five to one thousand two hundred and ninety-six.

A factor of more than five, at fixed resolution.

Same expressiveness, different resolution

The reverse case is sharper, and it is the sign on this page.

Three schemas permit exactly one thousand two hundred and ninety-six questions: four filters of five choices; five filters of two, two, three, five and five; and six filters of two, two, two, two, three and three.

Their cell counts are six hundred and twenty-five, three hundred, and one hundred and forty-four.

Schema	Cells	Questions
5·5·5·5	625	1296
2·2·3·5·5	300	1296
2·2·2·2·3·3	144	1296

The first resolves more than four times what the third resolves, while offering the reader exactly the same number of ways to ask.

Two boards can give identical answers to "how many questions can I put to this?" and differ by a factor of four in how much they can actually tell apart.

Why they diverge

Adding one to a small number changes it proportionally more than adding one to a large one.

A facet of six contributes a factor of six to the cells and seven to the questions — a ratio of seven sixths, about 1.17. A facet of two contributes two and three — a ratio of three halves, 1.5.

So a schema built from many small facets inflates its question count relative to its resolution, and a schema built from few large facets does the opposite. Both products use the same inputs; the two operations simply weight those inputs differently.

The extreme cases bound it. Six filters of two have sixty-four cells and seven hundred and twenty-nine questions: the ratio between them is more than eleven. One filter of six has six cells and seven questions: a ratio of about 1.17.

What a reader sees of the difference

Nothing reliable.

Four filters of five choices and six filters of mixed small sizes look like different boards, and a reader will register that one has more controls. What they will not register is that the four-filter board resolves more than four times as much.

The sidebar carries both figures implicitly and displays neither. Two multiplications recover them, and the alternative — counting controls — tracks the question count loosely and the resolution not at all.

One schema that sits in both largest classes

The last row of the table is worth a second look.

Six filters of two, two, two, two, three and three resolves one hundred and forty-four listings — putting it in the largest class of schemas sharing a cell count, which holds seven. It also permits one thousand two hundred and ninety-six questions — putting it in the largest class sharing a question count, which holds three.

So this one schema is maximally ambiguous on each figure taken alone. Quote its resolution and six other structures answer to the description. Quote its question count and two others do.

Quote both and it is identified uniquely, like every other schema in the record.

Where the divergence comes from, in one line

Resolution multiplies the sizes. Expressiveness multiplies the sizes each raised by one. Adding one to a two is a fifty per cent increase; adding one to a six is under seventeen.

So a schema built from small facets inflates the second product relative to the first, and a schema built from large ones does the reverse. Everything in this section is that one sentence, worked out on particular numbers.

What this means for comparing two boards

Comparing on one figure can rank them backwards.

Board A resolves six hundred and twenty-five kinds of listing and permits one thousand two hundred and ninety-six questions. Board B resolves one hundred and forty-four and permits the same one thousand two hundred and ninety-six. Judged on questions, they are equal. Judged on resolution, A is more than four times B.

Judged on the sidebar — the thing a reader actually sees — B looks richer: six controls against four.

All three readings are of the same pair of boards, and two of them are misleading. The only comparison that survives is the one that quotes both figures, and it takes two minutes to make.